The datasheet for the LM3915 tells you what the resistors do and how to calculate them. There are many circuits in the datasheet.
R3 is 1.5k in your circuit and it sets the current in the LEDs. The current is calculated with Ohm's Law and the LED currents are shown on a graph in the datasheet as 8mA which is fairly dim.
If the power supply voltage is much higher than the voltage of an LED, the current is high, there are no resistors in series with the LEDs, the mode is BAR and all LEDs are continuously lighted then the LM3915 will get too hot.
The datasheet says you can add one power resistor feeding power to all LEDs to share the heat which I did in the LM3915 that I use as a VU meter that has a 9V supply, the LED currents are very high to be bright and they are red LEDs with a low 1.8V voltage. I used a 10 ohms/1W resistor.
For R3 in your circuit I use 390 ohms so my LED currents are all 26mA so they are very bright. Then my LM3915 and the 10 ohms resistor get hot but not too hot.